Peckness of Edge Posets

For any graded poset P, we define a new graded poset, ??(P), whose elements are the edges in the Hasse diagram of P. For any group G acting on the boolean algebra B _n in a rank-preserving fashion we conjecture that ??(B _n /G) is Peck. We prove that the conjecture holds for “common cover transitive” actions. We give some infinite families of common cover transitive actions and show that the common cover transitive actions are closed under direct and semidirect products.     

On <Emphasis Type="Italic">n</Emphasis>-cubic Pyramid Algebras

In this paper we study a class of algebras having n-dimensional pyramid shaped quiver with n-cubic cells, which we called n-cubic pyramid algebras. This class of algebras includes the quadratic dual of the basic n-Auslander absolutely n-complete algebras introduced by Iyama. We show that the projective resolutions of the simples of n-cubic pyramid algebras can be characterized by n-cuboids, and prove that they are periodic. So these algebras are almost Koszul and (n?1)-translation algebras. We also recover Iyama’s cone construction for n-Auslander absolutely n-complete algebras using n-cubic pyramid algebras and the theory of n-translation algebras.     

The Leavitt Path Algebras of Generalized Cayley Graphs

Let n be a positive integer. For each \({0 \leq j \leq n-1}\), we let \({C_{n}^{j}}\) denote Cayley graph for the cyclic group \({\mathbb{Z}_n}\) with respect to the subset \({\{1, j\}}\). For any such pair (n, j), we compute the size of the Grothendieck group of the Leavitt path algebra \({L_K(C_{n}^{j})}\); the analysis is related to a collection of integer sequences described by Haselgrove in the 1940s. When j = 0, 1, or 2, we are able to extract enough additional information about the structure of these Grothendieck groups so that we may apply a Kirchberg-Phillips-type result to explicitly realize the algebras \({L_K(C_{n}^{j})}\) as the Leavitt path algebras of graphs having at most three vertices. The analysis in the j = 2 case leads us to some perhaps surprising and apparently nontrivial connections to the classical Fibonacci sequence.     

The Global Dimension of the full Transformation Monoid (with an Appendix by V. Mazorchuk and B. Steinberg)

The representation theory of the symmetric group has been intensively studied for over 100 years and is one of the gems of modern mathematics. The full transformation monoid \(\mathfrak {T}_{n}\) (the monoid of all self-maps of an n-element set) is the monoid analogue of the symmetric group. The investigation of its representation theory was begun by Hewitt and Zuckerman in 1957. Its character table was computed by Putcha in 1996 and its representation type was determined in a series of papers by Ponizovski?, Putcha and Ringel between 1987 and 2000. From their work, one can deduce that the global dimension of \(\mathbb {C}\mathfrak {T}_{n}\) is n?1 for n = 1, 2, 3, 4. We prove in this paper that the global dimension is n?1 for all n ≥ 1 and, moreover, we provide an explicit minimal projective resolution of the trivial module of length n?1. In an appendix with V. Mazorchuk we compute the indecomposable tilting modules of \(\mathbb {C}\mathfrak T_{n}\) with respect to Putcha’s quasi-hereditary structure and the Ringel dual (up to Morita equivalence).     

SPECIAL REDUCTIVE GROUPS OVER AN ARBITRARY FIELD

A linear algebraic group G defined over a field k is called special if every G-torsor over every field extension of k is trivial. In 1958 Grothendieck classified special groups in the case where the base field is algebraically closed. In this paper we describe the derived subgroup and the coradical of a special reductive group over an arbitrary field k. We also classify special semisimple groups, special reductive groups of inner type, and special quasisplit reductive groups over an arbitrary field k. Finally, we give an application to a conjecture of Serre.     

On Shalika models and <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-functions

We use modular symbols to construct p-adic L-functions for cohomological cuspidal automorphic representations on GL(2n), which admit a Shalika model. Our construction differs from former ones in that it systematically makes use of the representation theory of p-adic groups.     

Hopf Modules and Representations of Finite Wreath Products

For a finite group G and nonnegative integer n ≥ 0, one may consider the associated tower \(G \wr S_{n} := S_{n} \ltimes G^{n}\) of wreath product groups. Zelevinsky associated to such a tower the structure of a positive self-adjoint Hopf algebra (PSH-algebra) R(G) on the direct sum over integers n ≥ 0 of the Grothendieck groups K ₀(R e p?G?S _n ). In this paper, we study the interaction via induction and restriction of the PSH-algebras R(G) and R(H) associated to finite groups H ? G. A class of Hopf modules over PSH-algebras with a compatibility between the comultiplication and multiplication involving the Hopf k ^{t h} -power map arise naturally and are studied independently. We also give an explicit formula for the natural PSH-algebra morphisms R(H) → R(G) and R(G) → R(H) arising from induction and restriction. In an appendix, we consider a family of subgroups of wreath product groups analogous to the subgroups G(m, p, n) of the wreath product cyclotomic complex reflection groups G(m, 1, n).     

Towards a Supercharacter Theory of Parabolic Subgroups

A supercharacter theory is constructed for the parabolic subgroups of the group GL(n, F_q) with blocks of orders less or equal to two. The author formulated the hypotheses on construction of a supercharacter theory for an arbitrary parabolic subgroup in GL(n, F_q).     

About the spectrum of Nagata rings

In this paper, we deal with the Nagata ring R(n) in case R is obtained by a (T, I, D) construction. We characterize when R(n) is a strong S-domain and catenarian. This study allows us to provide several interesting applications and examples.     

Double-bosonization and Majid’s conjecture (IV): Type-crossings from A to BCD

Both in Majid's double-bosonization theory and in Rosso's quantum shuffle theory, the rankinductive and type-crossing construction for U_q(g)'s is still a remaining open question. In this paper, working in Majid's framework, based on the generalized double-bosonization theorem we proved before, we further describe explicitly the type-crossing construction of U_q(g)'s for(BCD)_n series directly from type An-1via adding a pair of dual braided groups determined by a pair of(R, R′)-matrices of type A derived from the respective suitably chosen representations. Combining with our results of the first three papers of this series, this solves Majid's conjecture, i.e., any quantum group U_q(g) associated to a simple Lie algebra g can be grown out of U_q(sl_2)recursively by a series of suitably chosen double-bosonization procedures.     

Consistent Digital Rays

Given a fixed origin o in the d-dimensional grid, we give a novel definition of digital rays dig(op) from o to each grid point p. Each digital ray dig(op) approximates the Euclidean line segment \(\overline {op}\) between o and p. The set of all digital rays satisfies a set of axioms analogous to the Euclidean axioms. We measure the approximation quality by the maximum Hausdorff distance between a digital ray and its Euclidean counterpart and establish an asymptotically tight Θ(log?n) bound in the n×n grid. The proof of the bound is based on discrepancy theory and a simple construction algorithm. Without a monotonicity property for digital rays the bound is improved to O(1). Digital rays enable us to define the family of digital star-shaped regions centered at o, which we use to design efficient algorithms for image processing problems.     

<Emphasis Type="Italic">n</Emphasis>-Abelian and <Emphasis Type="Italic">n</Emphasis>-exact categories

We introduce n-abelian and n-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that n-cluster-tilting subcategories of abelian (resp. exact) categories are n-abelian (resp. n-exact). These results allow to construct several examples of n-abelian and n-exact categories. Conversely, we prove that n-abelian categories satisfying certain mild assumptions can be realized as n-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius n-exact category has a natural \((n+2)\)-angulated structure in the sense of Geiß–Keller–Oppermann. We give several examples of n-abelian and n-exact categories which have appeared in representation theory, commutative algebra, commutative and non-commutative algebraic geometry.     

Phragmén-Lindelöf Theorems and p-harmonic Measures for Sets Near Low-dimensional Hyperplanes

We prove estimates of a p-harmonic measure, p∈(n?m,∞], for sets in Rⁿ which are close to an m-dimensional hyperplane Λ?Rⁿ, m∈[0,n?1]. Using these estimates, we derive results of Phragmén-Lindelöf type in unbounded domains Ω?Rⁿ?Λ for p-subharmonic functions. Moreover, we give local and global growth estimates for p-harmonic functions, vanishing on sets in Rⁿ, which are close to an m-dimensional hyperplane.     

Periods and nonvanishing of central <Emphasis Type="Italic">L</Emphasis>-values for GL(2<Emphasis Type="Italic">n</Emphasis>)

Let π be a cuspidal automorphic representation of PGL(2n) over a number field F, and η the quadratic idèle class character attached to a quadratic extension E/F. Guo and Jacquet conjectured a relation between the nonvanishing of L(1/2, π)L(1/2, π ? η) for π of symplectic type and the nonvanishing of certain GL(n,E) periods. When n = 1, this specializes to a well-known result of Waldspurger. We prove this conjecture, and related global results, under some local hypotheses using a simple relative trace formula.We then apply these global results to obtain local results on distinguished supercuspidal representations, which partially establish a conjecture of Prasad and Takloo-Bighash.     

Homological algebra i $${\varvec{}}$$-abelia categories

In this paper, we study the homological theory in n-abelian categories. First, we prove some useful properties of n-abelian categories, such as \((n+2)\times (n+2)\)-lemma, 5-lemma and n-Horseshoes lemma. Secondly, we introduce the notions of right(left) n-derived functors of left(right) n-exact functors, n-(co)resolutions, and n-homological dimensions of n-abelian categories. For an n-exact sequence, we show that the long n-exact sequence theorem holds as a generalization of the classical long exact sequence theorem. As a generalization of \(\textsf {Ext}^*(-,-)\), we study the n-derived functor \(\textsf {nExt}^*(-,-)\) of hom-functor \(\mathrm {Hom}(-,-)\). We give an isomorphism between the abelian group of equivalent classes of m-fold n-extensions \(\textsf {nE}^m(A,B)\) of A, B and \(\textsf {nExt}_{\mathcal A}^m(A,B)\) using n-Baer sum for \(m,n\ge 1\).     

Basics of the Quasiconformal Analysis of a Two-Index Scale of Spatial Mappings

We define a scale of mappings that depends on two real parameters p and q, n?1 ≤ q ≤ p < ∞ and a weight function θ. In the case of q = p = n, θ ≡ 1, we obtain the well-known mappings with bounded distortion. Mappings of a two-index scale inherit many properties of mappings with bounded distortion. They are used for solving a few problems of global analysis and applied problems.     

Normal number constructions for Cantor series with slowly growing bases

Let Q = (q_n)_n=1^∞ be a sequence of bases with q_i ≥ 2. In the case when the q_i are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose Q-Cantor series expansion is both Q-normal and Q-distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of Q, and from this construction we can provide computable constructions of numbers with atypical normality properties.